An Alternate to the Unified Distribution with Application to Breast Cancer Data
The Unified (θ) distribution appeared in , and in this short note we introduce a variant of this distribution. Further, a member of this family is shown to be significant in cancer modeling.
Bates distribution; Unified distribution; Cancer modeling
THE UNIFIED (θ) DISTRIBUTION
For x ∊(0,1) , the PDF is given by
Remark 3.1. For details on the derivation of this family, see 
THE NEW FAMILY
Let X be Bates on (0, 1) , and denote its PDF as f x (x,n) and consider
where k and ø are defined as in . Suppose Q=q, and consider =a+(b-a)J, where the random variable J has density q then
So, the CDF of Y is
By differentiation the PDF of Y is
So, if ø=1, then our density becomes
The class of all distributions with the above density is denoted A Unified (a, b, θ)
Remark 4.1. The families A Unified (a, b, θ) and Unified (θ) (of Section 1) coincide if and only if a=0 and b=1
THE UNIFIED (θ) GENERATED FAMILY OF DISTRIBUTIONS
The A Unified(a, b, θ) generated family of distributions has CDF
where 0 ≠θ ϵR, and K(x) is the CDF of some baseline distribution. Assuming the baseline distribution is Weibull with the following CDF
for x, g, h>0, then from the integral we have the following
Proposition 5.1. The CDF of the A Unified (0, 1, θ) Weibull family is given by
where x, g, h>0 and 0 ≠ ϵR
Obviously, the PDF can be obtained by differentiation. We write H AUW g h (θ, , ), if H is a A Unified (a, b, θ) Weibull random variable. The family in the above Proposition is a good fit to real life data as shown below. The MLE estimates are obtained using the software MATHEMATICA (Figure 1) .
In the present paper we introduced an alternate to the Unified distribution appearing in , and showed the Unified distribution is good in fitting real life data.
I am grateful to Oscar Alberto Quijano Acura of Concordia University, Montreal, Canada for pointing out an error in an earlier version of this paper.
- Oscar AQA (2019) The unified distribution. Journal of Statistical Distributions and Applications 6: 13.
- Bates distribution. Wikipedia, The Free Encyclopedia.
- Jayakumar K, Girish BM (2017) T-Transmuted X family of distributions. STATISTICA 77(3): 251-276.
Received date: November 19, 2019
Published date: November 25, 2019
Address for correspondence
Clement Boateng Ampadu; 31 Carrolton Road, Boston, MA 02132-6303, USA
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How to cite this article
Clement BA. An Alternate to the Unified Distribution with Application Breast Cancer Data. 2019 - 1(3) OAJBS.ID.000117.